Stochastic Models for Turbulenceqidi/filtering18/Lecture5.pdf · The Test Model Forced Turbulent...
Transcript of Stochastic Models for Turbulenceqidi/filtering18/Lecture5.pdf · The Test Model Forced Turbulent...
The Test Model Forced Turbulent Signals
Stochastic Models for TurbulenceMajda-Harlim Chapter 5
Mitch Bushuk & Themis Sapsis
February 16, 2011
The Test Model Forced Turbulent Signals
Outline
1. Motivation and a Test Model for Turbulence
2. Turbulent signals with Forcing and Dissipation
3. Statistics of Turbulent Solutions in Physical Space
4. Turbulent Rossby Waves
The Test Model Forced Turbulent Signals
Motivation
• Turbulent flows are highly irregular and need to be analyzed in
a statistical sense. This lends itself to the idea of modeling
turbulence as a stochastic process
• Coarse grained models are unable to resolve small scale
turbulence
• Large physical systems often have small scale turbulent
processes which are governed by unknown dynamics
We choose to parametrize resolved and unresolvedturbulence with spatially correlated white noise forcing
The Test Model Forced Turbulent Signals
A Stochastic Test Model for Turbulent Signals
Consider the initial value problem for the following scalar
stochastic PDE:
@u(x ,t)@t = P( @
@x )u(x , t)� �( @@x )u(x , t) + F (x , t) + �(x)W (t)
u(x , 0) = u0(x)
where,
• P( @@x ) is an operator constructed from odd derivatives
• �( @@x ) is an operator constructed from even derivatives
• F (x , t) is a deterministic forcing
• �(x)W (t) is spatially correlated white-noise forcing
• u0(x) ⇠ N(x ,�2)
The Test Model Forced Turbulent Signals
Test Model Operators
The operators satisfy
P( @@x )e
ikx= p(ik)e ikx
�( @@x )e
ikx= �(ik)e ikx
Assume p(ik) is wave-like:
p(ik) = i!k
where �!k is the dispersion relation.
�(ik) is the damping operator and satisfies
�(ik) > 0 8 k 6= 0
The Test Model Forced Turbulent Signals
The Stochastically Forced Dissipative Advection Equation
Taking P( @@x ) = �c @
@x and �( @@x ) = �d + µ @2
@x2 , we have,
@u(x ,t)@t = �c @u(x ,t)
@x � du(x , t) + µ@2u(x ,t)@x2 + F (x , t) + �(x)W (t)
• p(ik) = i!k = �ick
• �(ik) = d + µk2
We seek a Fourier series solution:
u(x , t) =1X
k=�1uk(t)e
ikx
where u�k = u⇤k
The Test Model Forced Turbulent Signals
Fourier Series Solution
@u(x ,t)@t = P( @
@x )u(x , t)� �( @@x )u(x , t) + F (x , t) + �(x)W (t)
Each uk(t) satisfies the SDE:
duk(t) = [p(ik)� �(ik)]uk(t)dt + Fk(t)dt + �kdWk(t)
Multiplying by the integrating factor e(�(ik)�p(ik))t, we have,
d(e(�(ik)�p(ik))t uk(t)) = e(�(ik)�p(ik))t(Fk(t)dt + �kdWk(t))
e(�(ik)�p(ik))t uk(t)� uk(0) =R t0 e(�(ik)�p(ik))s Fk(s)ds + �k
R t0 e(�(ik)�p(ik))sdWk(s)
Thus,
uk(t) = uk(0)e(p(ik)��(ik))t+R t0 e(�(ik)�p(ik))(s�t)Fk(s)ds +
�kR t0 e(�(ik)�p(ik))(s�t)dWk(s)
The Test Model Forced Turbulent Signals
Large Time Behaviour
Taking F (x , t) = 0, we have
uk(t) = uk(0)e(p(ik)��(ik))t+ �k
R t0 e(�(ik)�p(ik))(s�t)dWk(s)
Note:Wk =W1+iW2p
2
E[uk(t)] = uk(0)e(p(ik)��(ik))t ! 0
E[uk(t)uk(t)⇤] = uk(0)uk(0)⇤e�2�(ik)t+
�k�k⇤2�(ik)(1� e�2�(ik)t
) !�2k
2�(ik)
We define the energy spectrum,
Ek =�2k
2�(ik) 1 k < 1
The Test Model Forced Turbulent Signals
Autocorrelation Function
Let �(ik) = �(ik)� p(ik)
Note that: �⇤(ik) = �(ik) + p(ik)
Rk(t, t + ⌧) = E[(uk(t)� ¯uk)(uk(t + ⌧)� ¯uk)]
= E[(�kR t0 e(�(ik)(s�t)dWk(s))(�k
R t+⌧0 e(�(ik)(s
0�t�⌧)dWk(s 0))⇤]
=
�2ke
��(ik)(2t+⌧)�p(ik)⌧R t0
R t+⌧0 e�(ik)s+�⇤(ik)s0 1
2E[dW1(s)dW1(s 0) +dW2(s)dW2(s 0)]
= �2ke
��(ik)(2t+⌧)�p(ik)⌧R t0
R t+⌧0 e�(ik)s+�⇤(ik)s0�(s � s 0)dsds 0
= �2ke
��(ik)(2t+⌧)�p(ik)⌧R t0 e�(ik)s+�⇤(ik)sds
= �2ke
��(ik)(2t+⌧)�p(ik)⌧ 12�(ik)(e
2�(ik)t � 1)
=�2k
2�(ik)e��(ik)(2t+⌧)�p(ik)⌧
(e2�(ik)t � 1)
=�2k
2�(ik)e��(ik)⌧�p(ik)⌧
(1� e�2�(ik)t)
The Test Model Forced Turbulent Signals
Autocorrelation Function
Finally, we have,
R(t, t + ⌧) =�2k
2�(ik)e��(ik)⌧�p(ik)⌧
(1� e�2�(ik)t)
In the large t limit,
R(t, t + ⌧) =�2k
2�(ik)e��(ik)⌧�p(ik)⌧
Thus,
Real(R(t, t + ⌧)) =�2k
2�(ik)e��(ik)⌧
cos(!k⌧) = Eke��(ik)⌧cos(!k⌧)
Decorrelation Time:1
�(ik)
The Test Model Forced Turbulent Signals
Calibrating the Noise Level
From observations, we can roughly determine the energy spectrum
and decorrelation time at each wavenumber.
Recall that Ek =�2k
2�(ik) . Thus, we can produce an estimate of the
noise level using
�k =p2�(ik)Ek
A typical turbulent energy spectra has the power law form:
Ek = E0|k |��
For example, if �(ik) = d + µk2,
�k = E 1/20 |k |��/2
(d + µk2)1/2
• � > 2 ) decreasing noise at small spatial scales
• � < 2 ) increasing noise at small spatial scales
The Test Model Forced Turbulent Signals
Damped Forced Solutions
Consider a forcing of the following form:
Fk(t) =
⇢Ae i!0(k)t k M0 k > M
The mean dynamics satisfy:
¯uk(t) = ¯uk(0)e��(ik)t+R t0 e�(ik)(s�t)Fk(s)ds
=
(¯uk(0)e��(ik)t
+Aei!0(k)t
�(ik)+i(!0(k)�!k )(1� e(��(ik)�!0(k))t) k M
¯uk(0)e��(ik)t k > M
The Test Model Forced Turbulent Signals
Test Problem with Resonant Forcing
We choose a resonant forcing: !0(k) = !k 8 k such that |k | M
Numerical integrations were performed on:
@u(x ,t)@t = �c @u(x ,t)
@x � du(x , t) + µ@2u(x ,t)@x2 + F (x , t) + �(x)W (t)
with parameters:
• c = 1
• d = 0
• µ = 0.01
• A = 0.1
• M = 20
• kmax = 61
• Ek = 1, Ek = k�5/3
• �k =p0.02k , �k =
p0.02k1/6
The Test Model Forced Turbulent Signals
Test Problem with Resonant Forcing
Sta$s$cs'of'Turbulent'solu$ons'in'physical'space'
We'have'the'solu$on'
Sta$s$cal'behavior'7'Mean'
'''''
'''''
Turbulent'Rossby'waves'Barotropic'Rossby'waves'with'phase'varying'only'in'the'north7south'direc$on'
known'from'observa$ons'that'on'scales'of'order'of'thousands'of'kilometers'these'waves'have'a'k−3'energy'spectrum'
Chapter 6:Filtering Turbulent Signals: Plentiful Observations
In particular, in this simplified context, we address the basic issues outlined in 1.a)-1.d) of Chapter 1.
6.1 A Mathematical Theory for Fourier Filter Reduction
6.1 A Mathematical Theory for Fourier Filter Reduction
6.1 A Mathematical Theory for Fourier Filter Reduction
6.1 A Mathematical Theory for Fourier Filter Reduction
6.1 A Mathematical Theory for Fourier Filter Reduction
6.1 A Mathematical Theory for Fourier Filter Reduction
6.1 A Mathematical Theory for Fourier Filter Reduction
6.1 The Number of Observation Points Equals the Number of Discrete MeshPoints:MathematicalTheory
6.1 The Number of Observation Points Equals the Number of Discrete MeshPoints:MathematicalTheory
6.1 The Number of Observation Points Equals the Number of Discrete MeshPoints:MathematicalTheory
6.1 The Number of Observation Points Equals the Number of Discrete MeshPoints:MathematicalTheory
6.1 The Number of Observation Points Equals the Number of Discrete MeshPoints:MathematicalTheory